Marta B. Rosales

Free vibrations of a spinning uniform beam with ends elastically restrained against rotation

The title problem is considered in the case where the spinning beam cross-section possesses only one axis of symmetry. An exact solution is obtained by means of an extension of Bauers approach, and with account taken of orthotropic properties of the beam supports. The effect of coupling with torsional modes is not taken into account in the present investigation.

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

Abstract The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.

A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium

This paper deals with the determination of the fundamental frequencies of Timoshenko beams in a Winkler-Pasternak medium by means of the variant of Rayleighs method which allows an optimization of the approximate modal functions through a non-integer exponential parameter. The case of a Timoshenko beam gives rise to a two-variable problem which constitutes an extension of the title method with respect to previous work. A static functional relationship between the two unknown functions is proposed. Numerical results are given for simply supported and clamped-clamped beams of uniform and variable cross section. Comparisons with available exact solutions were made, showing very good agreement.

Vibration of orthotropic plates: discussion on the completeness of the solutions used in direct methods

Fil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingenieria; Argentina

Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.

Alternative Approach for the Exact Solution of the Forced Vibrations of Beams

The present work is an extension of a tool vastly used by the authors to solve static boundary problems in one, two, and even three dimensions. It consists in a so-called generalized solution with special trigonometric Fourier functions to solve the equations of motion of beams. An important theorem that guarantees that the classic answer is attained through an alternative way is demonstrated. In other words, it is a variational methodology to solve differential equations in engineering. An example solved numerically completes the present proposal.

Free flexural-torsional vibrations of a uniform spinning beam

Abstract The present study deals with the free vibration coupling of bending and torsion of a uniform spinning beam. The exact solution is presented. An example is carried out numerically in order to point out the influence of whole coupling. Comparison is made with previous works on bending without torsional coupling and torsion neglecting bending coupling. It is concluded that flexural-torsional coupling is unavoidable. In effect, in neglecting it some fundamental frequencies are lost and others are in error.

Deflections in sawn timber beams with stochastic properties

A stochastic model is proposed to study the behavior of structural sawn beams of Argentinean Eucalyptusgrandis with the aim of improving the predictability of the elastic deformations. The enhancement of the mid-span deflection calculation is based on a probabilistic model of the Modulus of Elasticity (MOE) and the representation of its lengthwise variability through a random field. The standard model that uses a MOE variable assumed random from piece to piece but deterministic (constant) within each piece is obtained as a particular case. In order to obtain a statistical representation of the MOE, the Principle of Maximum Entropy (PME) is employed. Experimental data obtained from bending tests are employed to find the parameters of the derived Probability Density Function (PDF). The PDF of the mid-span deformations is numerically obtained through the Stochastic Finite Element Method (SFEM) and Monte Carlo Simulations (MCS). Numerical results are validated with experimental values. Deflections of structural sized beams under usual loads are obtained. Finally, the stochastic model is used to compare with the serviceability requirements established in the Argentinean design code. It is shown that the structural performance of timber beams is found through a more realistic material approach.

Eigenproblems in timber structural elements with uncertain properties

Abstract A stochastic model of sawn timber structural elements of Argentinean Eucalyptusgrandis is applied to the study of two eigenproblems. One is the free vibrations problem which, after being solved, yields the natural frequencies and modes. The other problem is the buckling of columns. Its solution leads to the buckling loads and modes. The governing differential equations are stated starting from the Euler–Bernoulli beam theory. Then, they are discretized and numerically approximated through the finite element method. The stochasticity is given by the mechanical properties involved in each problem. The lengthwise variabilities of the modulus of elasticity and of the second moment of the cross-sectional area are simulated to account for the presence of knots. The variability of the mass density among structural elements is also considered. The statistics of the solutions are obtained. The probability density functions of the natural frequencies and the buckling loads are numerically obtained through a stochastic finite element concept using Monte Carlo simulation. Numerical results for the first natural frequency are validated with experimental values. The mode shape statistics are also analyzed. Frequently the presence of knots in sawn timber structures is disregarded, usually due to the lack of data and the availability of an adequate representation. The model herein presented contributes to attain a more realistic description of structures made out of sawn timber due to the unavoidable variability of the properties, in particular the presence of knots .

A generalization of the membrane-plate analogy to non-homogeneous polygonal domains consisting of homogeneous subdomains

The well-known membrane-plate analogy that relates the natural frequencies when dealing with polygonal homogeneous domains is herein extended to non-homogeneous systems comprised of homogeneous subdomains. The analogy is generalized and demonstrated and it is shown that certain restrictions among the frequency parameters of the membranes and plates arise. Several examples of membranes and plates with interfaces separating areas with different material properties are numerically solved with different approaches. The subdomains are separated by straight, curved, and closed line interfaces. It is shown that the analogy is verified provided that the restrictions are satisfied. The analogy is first demonstrated and presented as a practical methodology to find the natural frequencies of membranes knowing the corresponding ones of the plates or vice versa. Second, the plate and membrane vibration problems, governed by the bi-Laplacian and Laplacian differential operators, respectively, can be solved without distinction, though under certain conditions, i.e., solve one of them and deduce the other using the analogy. Various numerical examples validate the analogy.